We have attributed the ill-conditioning observed above as \(\varepsilon \rightarrow 0\)  to the use of P1 elements. To test if this explanation is
plausible we repeated this test with P2 elements. Since this finite element
space contains a non-trivial divergence free subspace, [1]}, our
intuition is that ill-conditioning would be reduced. Table 3 below presents \(Est.3\simeq \varepsilon ^{-1}h^{-1}\frac{||\nabla \cdot u^{\varepsilon }||}{||u^{\varepsilon }||}\)  values. \(Est.2\simeq ||f^{h}||/||u^{\varepsilon }||\) 
values, not presented, were significantly smaller than \(Est.3\)  and converged
to \(1.13\)  as \(\varepsilon ,h\rightarrow 0\) .
\(\begin{array}{c|cccccc}{\small \varepsilon \Downarrow \setminus h\Rightarrow } & {\small 0.125} &{\small 6.2E-2} & {\small 3.1E-2} & \text{ }{\small 1.6E-2} & {\small 7.8E-3}& {\small 3.9E-3} \\\hline {\small 1} & {\small 25} & {\small 50} & {\small 1.0E2} & {\small 2.0E2} &{\small 4.0E2} & {\small 8.0E2} \\{\small 6.3E-2} & {\small 3.1E2} & {\small 6.2E2} & {\small 1.2E3} & {\small 2.5E3} & {\small 5.0E3} & {\small 1.0E4} \\{\small 3.9E-3} & {\small 6.7E2} & {\small 1.3E3} & {\small 2.5E3} & {\small 5.0E3} & {\small 1.0E4} & {\small 2.0E4} \\{\small 2.4E-4} & {\small 7.1E2} & {\small 1.3E3} & {\small 2.6E3} & {\small 5.1E3} & {\small 1.0E4} & {\small 2.0E4} \\{\small 1.5E-5} & {\small 7.1E2} & {\small 1.3E3} & {\small 2.6E3} & {\small 5.2E3} & {\small 1.0E4} & {\small 2.0E4} \\{\small 9.5E-7} & {\small 7.1E2} & {\small 1.3E3} & {\small 2.7E3} & {\small 5.3E3} & {\small 1.1E4} & {\small 2.1E4} \\{\small 6.0E-8} & {\small 7.1E2} & {\small 1.3E3} & {\small 2.7E3} & {\small 5.3E3} & {\small 1.1E4} & {\small 2.1E4}\end{array}\\\text{Table 3: Values of }Est.3\simeq \varepsilon ^{-1}h^{-1}\frac{||\nabla \cdot u^{\varepsilon }||}{||u^{\varepsilon }||}\text{, P2 elements}\) 
